We formulate standard and multilevel Monte Carlo methods for the $k$th moment $\mathbb{M}^k_\varepsilon[\xi]$ of a Banach space valued random variable $\xi\colon\Omega\to E$, interpreted as an element of the $k$-fold injective tensor product space $\otimes^k_\varepsilon E$. For the standard Monte Carlo estimator of $\mathbb{M}^k_\varepsilon[\xi]$, we prove the $k$-independent convergence rate $1-\frac{1}{p}$ in the $L_q(\Omega;\otimes^k_\varepsilon E)$-norm, provided that (i) $\xi\in L_{kq}(\Omega;E)$ and (ii) $q\in[p,\infty)$, where $p\in[1,2]$ is the Rademacher type of $E$. By using the fact that Rademacher averages are dominated by Gaussian sums combined with a version of Slepian's inequality for Gaussian processes due to Fernique, we moreover derive corresponding results for multilevel Monte Carlo methods, including a rigorous error estimate in the $L_q(\Omega;\otimes^k_\varepsilon E)$-norm and the optimization of the computational cost for a given accuracy. Whenever the type of the Banach space $E$ is $p=2$, our findings coincide with known results for Hilbert space valued random variables. We illustrate the abstract results by three model problems: second-order elliptic PDEs with random forcing or random coefficient, and stochastic evolution equations. In these cases, the solution processes naturally take values in non-Hilbertian Banach spaces. Further applications, where physical modeling constraints impose a setting in Banach spaces of type $p<2$, are indicated.

翻译：我们针对Banach空间值随机变量$\xi\colon\Omega\to E$的$k$阶矩$\mathbb{M}^k_\varepsilon[\xi]$，将其解释为$k$重单射张量积空间$\otimes^k_\varepsilon E$中的元素，构建了标准蒙特卡洛方法和多层蒙特卡洛方法。对于$\mathbb{M}^k_\varepsilon[\xi]$的标准蒙特卡洛估计量，我们证明了在$L_q(\Omega;\otimes^k_\varepsilon E)$-范数下收敛速率为$1-\frac{1}{p}$且与$k$无关，前提是(i) $\xi\in L_{kq}(\Omega;E)$ 且(ii) $q\in[p,\infty)$，其中$p\in[1,2]$为$E$的Rademacher型。利用Rademacher平均由高斯和支配的性质，结合Fernique关于高斯过程的Slepian不等式的一个变体，我们进一步推导出多层蒙特卡洛方法的相应结果，包括$L_q(\Omega;\otimes^k_\varepsilon E)$-范数下的严格误差估计以及给定精度下计算成本的优化。当Banach空间$E$的型$p=2$时，我们的结论与Hilbert空间值随机变量的已知结果一致。我们通过三个模型问题阐明这些抽象结果：具有随机强迫项或随机系数的二阶椭圆型偏微分方程，以及随机发展方程。在这些情形中，解过程自然取值于非Hilbert型Banach空间。文中还指出了其他应用场景，其中物理建模约束需设定在型$p<2$的Banach空间中。